
In addition, analogously, we can define the components of the discrete equilibrium central moments of the regular square lattice as \(\eta ^{c,eqr}_{x^m y^n}\)  as part of the vector \(\mathbf {m}^{c,eqr}\) . They can be obtained by matching the corresponding continuous central moments of the Maxwell distribution function [1]}, [2]}, and the components of the central moments of the sources can be prescribed
to yield consistency with the NS equations [2]}. Thus,
\(\mathbf {m}^{c,eqr}&=& \left(\eta _0^{c,eqr}, \eta _x^{c,eqr}, \eta _y^{c,eqr}, \eta _{xx}^{c,eqr}, \eta _{yy}^{c,eqr}, \eta _{xy}^{c,eqr} , \eta _{xxy}^{c,eqr}, \eta _{xyy}^{c,eqr}, \eta _{xxyy}^{c,eqr}\right)^{\dag },\nonumber \\&=&\left(0, 0, 0,\rho c_s^2, \rho c_s^2, 0, 0, 0, \rho c_s^4 \right)^{\dag }, \\\mathbf {\Phi }^{c}&=& \left(\sigma _0^{c}, \sigma _x^{c}, \sigma _y^{c}, \sigma _{xx}^{c}, \sigma _{yy}^{c},\sigma _{xy}^{c}, \sigma _{xxy}^{c}, \sigma _{xyy}^{c}, \sigma _{xxyy}^{c}\right)^{\dag },\nonumber \\&=&\left(0, F_x, F_y, 0, 0, 0, 0, 0, 0 \right)^{\dag }.\) 
